Crystal Bases

Representations and Combinatorics

Daniel Bump, Anne Schilling

• Publisher: WSPC
• Year: 2017
• ISBN:
  9789814733465
• Topic: Mathématiques
• Subtopic: Comptage et numération

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This unique book provides the first introduction to crystal base theory from the combinatorial point of view. Crystal base theory was developed by Kashiwara and Lusztig from the perspective of quantum groups. Its power comes from the fact that it addresses many questions in representation theory and mathematical physics by combinatorial means. This book approaches the subject directly from combinatorics, building crystals through local axioms (based on ideas by Stembridge) and virtual crystals. It also emphasizes parallels between the representation theory of the symmetric and general linear groups and phenomena in combinatorics. The combinatorial approach is linked to representation theory through the analysis of Demazure crystals. The relationship of crystals to tropical geometry is also explained.

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Contents: Introduction;Kashiwara Crystals;Crystals of Tableaux;Stembridge Crystals;Virtual, Fundamental, and Normal Crystals;Crystals of Tableaux II;Insertion Algorithms;The Plactic Monoid;Bicrystals and the Littlewood–Richardson Rule;Crystals for Stanley Symmetric Functions;Patterns and the Weyl Group Action;The β ∞ Crystal;Demazure Crystals;The ⋆-Involution of β ∞;Crystals and Tropical Geometry;Further Topics; --> -->
Readership: Graduate students and researchers interested in understanding from a viewpoint of combinatorics on crystal base theory.
-->Combinatorics, Representation Theory, Open-Source Mathematical Software System Sage

• First textbook that approaches crystal base theory solely from the combinatorial perspective
• The presentation uses the Stembridge local axioms and virtual crystals to uniquely characterize classical crystals
• The textbook incorporates examples on how to compute and experiment with crystals using the open-source software system Sage

Chapter 1

Introduction

Crystal bases or Kashiwara crystals are combinatorial structures that mirror representations of Lie groups. Historically, crystal bases were developed independently around 1990 from two independent sources.

On the one hand, [Kashiwara (1990, 1991, 1994)] showed that modules of quantum groups have “crystal bases” with remarkable combinatorial properties. Independently, [Lusztig (1990a, b)] introduced canonical bases from a more geometric perspective. Quantum groups are Hopf algebras that are “noncommutative” analogs of Lie groups. A particular class of quantum groups, quantized enveloping algebras, are deformations (in the category of Hopf algebras) of the universal enveloping algebras of Lie groups. They were described independently by [Drinfel′d (1985)] and [Jimbo (1985)] to explain developments in mathematical physics. Every representation of the Lie group gives rise to a representation of the corresponding quantized enveloping algebra, and Kashiwara showed that these modules have crystal bases whose properties he axiomatized and proved, using deep methods from quantum groups.

On the other hand, crystals also came about through the analysis of [Littelmann (1994, 1995b)] of standard monomial theory ([Lakshmibai, Musili and Seshadri (1979); Lakshmibai and Seshadri (1991)]). Borel and Weil and later [Bott (1957)] showed that representations of Lie groups can be realized as sections of line bundles on flag varieties. [Demazure (1974, 1976)] had found additional structure in these modules. Inspired by work of [Hodge (1943)] on the cohomology of Grassmannians, Seshadri and Lakshmibai found convenient bases of these modules of sections that are indexed by tableaux. Peter Littelmann, in the early 1990’s, reinterpreted these bases as paths through a vector space containing the weight lattice and showed that they may be organized into crystals like those found by Kashiwara in the theory of quantum groups. [Kashiwara (1996); Joseph (1995)] then proved that the crystals arising from quantum groups are the same as the crystals arising from the Littelmann paths.

In retrospect, some older work in the combinatorics of tableaux can be understood in terms of crystals. [Littlewood (1940)] showed that a Schur polynomial, which is the character of an irreducible representation of GL(n), had a combinatorial definition as a sum over tableaux. In fact, both the irreducible representations of the symmetric group and the general linear group were known to have bases indexed by tableaux, and the Robinson–Schensted–Knuth (RSK) algorithm [Knuth (1970, 1998)] gave bijections that are combinatorial analogs of certain isomorphisms between such modules of Lie groups and the symmetric groups. Later, [Lascoux and Schützenberger (1981)] gave a multiplicative structure on the set of tableaux, called the plactic monoid, that is closely related to RSK. All of these topics fit into the theory of crystal bases and the connections will be discussed in Chapters 7 and 8.

Crystals appear in many other contexts from mathematical physics and combinatorics to number theory. We will not attempt to survey all of these here.

In this book, we will limit ourselves to crystals associated to finite-dimensional Lie algebras, omitting the important topic of crystals of representations of infinite-dimensional Lie algebras. Within this limited scope, we have tried to prove the essential facts using combinatorial methods. The facts one wants to prove are as follows.

Given a reductive complex Lie group G, there is an associated weight lattice Λ with a cone of dominant weights. Given a dominant weight λ, there is a unique irreducible representation of highest weight λ. There are two operations on these that we are particularly concerned with: tensor product of representations and branching, or restriction, to Levi subgroups.

In the theory of crystal bases, one starts with the same weight lattice and cone of dominant weights. Instead of a representation, one would like to associate a special crystal to each dominant weight. If the representation is irreducible, the crystal should be connected. There may be many connected crystals with a given highest weight, but it turns out that there is one particular one that we call normal. We think of this as the “crystal of the representation.” More generally, a crystal that is the disjoint union of such crystals, is to be considered normal.

The operations of tensor product and Levi branching from representation theory also make sense for crystals. The usefulness of the class of normal crystals is that the decomposition of a crystal into irreducibles with respect to these operations is again normal. Moreover, the decomposition of a representation obtained by tensoring representations or branching a represention to a Levi subgroup gives the same multiplicities as the decomposition of the tensor product or Levi branching of the corresponding normal crystals into irreducibles.

There are several ways of defining normal crystals. [Kashiwara (1990, 1991, 1994)] and [Littelmann (1994, 1995b)] gave two different definitions, which then were shown to be equivalent. We give yet another definition of normal crystals, based on two key ideas: Stembridge crystals ([Stembridge (2003)]) and virtual crystals ([Kashiwara (1996); Baker (2000)]). For the simply-laced Cartan types, [Stembridge (2003)] showed how to characterize the normal crystals axiomatically. This is subject of Chapter 4. This approach does not work as well for the non-simply-laced types, but for these, there is a way of embedding certain crystals into crystals of corresponding simply-laced types. For example, to construct a normal Sp(2r) crystal (for the non-simply-laced Cartan type C_r) first one constructs a GL(2r) crystal (for the simply-laced Cartan type A_2r−1). Then one finds the symplectic crystal as a “virtual crystal” inside the GL(2r) one. This way, one may reduce many problems about crystals to the simply-laced case, including the construction of the normal crystals (see Chapter 5).

Once the normal crystals are constructed, we have a bijection between normal crystals and finite-dimensional representations of the Lie group G. In this bijection connected crystals correspond to irreducible representations. A representation has a character, and so does a crystal. If we can show that the character of the crystal equals the character of the representation, then it will follow that the decomposition of a tensor product of crystals into irreducibles has the same multiplicities as the corresponding tensor product of representations; and similarly the Levi branching rules for crystals and representations will be the same.

How does one prove that the character of an irreducible representation is the same as that of the corresponding normal crystal? The approach we follow depends on the Demazure character formula, which constructs the character of a representation in stages. In the Demazure character formula, given a dominant weight λ and a Weyl group element w, there is a Demazure character ∂_w(t^λ). This generalizes the character χ_λ of the irreducible representation with highest weight λ, because if w equals the long Weyl group element w₀, then ∂_w0(t^λ) = χ_λ. For general w, the Demazure character ∂_w(t^λ) is a part of χ_λ.

The corresponding construction in crystals, due to [Littelmann (1995a)] and [Kashiwara (1993)], constructs certain subsets B_λ(w) of the connected normal crystal B_λ with highest weight λ. The main fact to be proved inductively is that the Demazure characters ∂_w(t^λ) associated with any Weyl group element w agrees with the character of the Demazure crystal B(w). This fact is called the refined Demazure character formula.

A direct approach to the refined Demazure character formula seems difficult, even in the simply-laced case armed with the Stembridge axioms. Instead, what works is to construct the Demazure crystals inside the crystal B_∞. This is an infinite crystal that contains a copy of B_λ for every dominant weight λ. We construct the Demazure crystals in B_∞ in Chapter 12 and then deduce the properties of their counterparts in B_λ. After this, we are able to finish the proof of the Demazure character formula for crystals, and thereby establish the relationship with representations as shown in Chapter 13.

The infinite crystal B_∞ is itself a remarkable combinatorial object. It is, in a sense, the crystal of a representation, albeit an infinite-dimensional one, the Verma module with weight 0. As we discover in the proof of the refined Demazure character formula, a clear understanding of B_∞ may be the key to the finite normal crystals. Therefore after we prove the Demazure character formula we investigate B_∞ in more depth. Chapter...